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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Two dimensional elliptic equation with critical nonlinear growth
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by Takayoshi Ogawa and Takashi Suzuki PDF
Trans. Amer. Math. Soc. 350 (1998), 4897-4918 Request permission


We study the asymptotic behavior of solutions to a semilinear elliptic equation associated with the critical nonlinear growth in two dimensions. \begin{equation*}\tag {1.1} \begin {cases} -\Delta u= \lambda ue^{u^2}, u>0 & \text {in $\Omega $}, \\ u=0 & \text {on $\partial \Omega $}, \end{cases} \end{equation*} where $\Omega$ is a unit disk in $\mathbb {R}^2$ and $\lambda$ denotes a positive parameter. We show that for a radially symmetric solution of (1.1) satisfies \begin{equation*} \int _{D}\left \vert \nabla u\right \vert ^{2}dx\rightarrow 4\pi ,\quad \lambda \searrow 0. \end{equation*} Moreover, by using the Pohozaev identity to the rescaled equation, we show that for any finite energy radially symmetric solutions to (1.1), there is a rescaled asymptotics such as \begin{equation*} u_m^2(\gamma _m x)-u_m^2 (\gamma _m)\to 2\log \frac {2}{1+|x|^2} \quad \text {as }\lambda _m\searrow 0 \end{equation*} locally uniformly in $x\in \mathbb R^2$. We also show some extensions of the above results for general two dimensional domains.
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Additional Information
  • Takayoshi Ogawa
  • Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93106
  • Address at time of publication: Graduate School of Mathematics, Kyushu University 36, Fukuoka, 812-8581, Japan
  • MR Author ID: 289654
  • Email:
  • Takashi Suzuki
  • Affiliation: Department of Mathematics, Osaka University, Toyonaka, Osaka 560, Japan
  • MR Author ID: 199324
  • Email:
  • Received by editor(s): January 29, 1996
  • Additional Notes: The first author is on long-term leave from the Graduate School of Polymathematics, Nagoya University, Nagoya 464-01 Japan.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 4897-4918
  • MSC (1991): Primary 35J60, 35P30, 35J20
  • DOI:
  • MathSciNet review: 1641254